Elliptic systems with the doubly critical exponent have been investigated by many authors (c.f. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]} and the references therein). However, much less is known when the \(L^2\) -norms \(\Vert u\Vert _2,\Vert v\Vert _2\)  are prescribed. Recently, Li, Zou in [8]} and Mederski, Schino in [9]} considered normalized solutions to coupled Schrödinger systems with a wide class of nonlinearities including the Sobolev critical terms. Here, we have to emphasize that the system considered by Li and Zou in [8]} contains only a single critical nonlinearity, and coupled terms of the systems considered by Mederski and Schino in [9]} must be Sobolev subcritical.
In this paper, we study the existence of normalized solutions to problem (REF ) with doubly Sobolev critical exponent involving coupling terms.
